Interplay of electron-phonon interaction and electron correlation in high-temperature superconductivity

We study the electron-phonon interaction in the strongly correlated superconducting cuprates. Two types of the electron-phonon interactions are introduced in the $t-J$ model; the diagonal and off-diagonal interactions which modify the formation energy of the Zhang-Rice singlet and its transfer integral, respectively. The characteristic phonon-momentum $\left(\overrightarrow{q}\right)$ and electron-momentum $\left(\overrightarrow{k}\right)$ dependence that resulted from the off-diagonal coupling can explain a variety of experiments. The vertex correction for the electron-phonon interaction is formulated in the SU(2) slave-boson theory by taking into account the collective modes in the superconducting ground states. It is shown that the vertex correction enhances the attractive potential for the d-wave paring mediated by phonon with $\overrightarrow{q}=\left[\pi \right(1\mathrm{}-\delta ),0\mathrm{}]$ around $\delta \cong 0.3,$ which corresponds to the half-breathing mode of the oxygen motion.